We further develop the asymptotic analytic approach to the study of scattering diagrams. We do so by analyzing the asymptotic behavior of Maurer--Cartan elements of a differential graded Lie algebra constructed from a (not-necessarily tropical) monoid-graded Lie algebra. In this framework, we give alternative differential geometric proofs of the consistent completion of scattering diagrams, originally proved by Kontsevich--Soibelman, Gross--Siebert and Bridgeland. We also give a geometric interpretation of theta functions and their wall-crossing. In the tropical setting, we interpret Maurer--Cartan elements, and therefore consistent scattering diagrams, in terms of the refined counting of tropical disks. We also describe theta functions, in both their tropical and Hall algebraic settings, in terms of distinguished flat sections of the Maurer--Cartan-deformed differential. In particular, this allows us to give a combinatorial description of Hall algebra theta functions for acyclic quivers with non-degenerate skew-symmetrized Euler forms.

In this paper, we derive the CR Reilly'’s formula and its applications to studying of the fi…rst eigenvalue estimate for CR Dirichlet eigenvalue problem and embedded p-minimal hypersurfaces. In particular, we obtain the …first Dirichlet eigenvalue estimate in a compact pseudo-hermitian manifold with boundary and the fi…rst eigenvalue estimate of the tangential sublaplacian on closed oriented embedded p-minimal hypersurfaces in a closed pseudohermitian manifold of vanishing torsion.

An inequality relating the Euler characteristic, the signature and the <i>L</i>₂-norm of the curvature of the bundle of densities is proved for a four-dimensional compact Einstein-Weyl manifold. This generalises the Hitchin-Thorpe inequality for Einstein manifolds. The case where equality occurs is discussed and related to Hitchin's classification of Ricci-flat self-dual four-manifolds and to the recent work of Gauduchon on closed non-exact Einstein-Weyl geometries.

We consider the inverse curvature flow of strictly convex hypersurfaces in the space form N of constant sectional curvature K_N with speed given by F^{−α}, where α∈(0,1] for K_N=0,−1 and α=1 for K_N=1, F is a smooth, symmetric homogeneous of degree one function which is inverse concave and has dual F_∗ approaching zero on the boundary of the positive cone Γ_+. We show that the ratio of the largest principal curvature to the smallest principal curvature of the flow hypersurface is controlled by its initial value. This can be used to prove the smooth convergence of the flows.

This note surveys and compares results on the separation of variables construction for soliton solutions of curvature equations including the K\"ahler-Ricci flow and the Lagrangian mean curvature flow. In the last section, we propose some new generalizations in the Lagrangian mean curvature flow case.